In this section I use the marginal rate of substitution in equation (5a) to determine the price of the canonical asset. As will become evident, the price of the canonical asset can be expressed as
The Canonical Asset
Rather than proceed with separate derivations for the prices and rates of return for different assets such as equity, short-term bills and long-term bonds, I will introduce a canonical asset that includes all of these assets as special cases. The canonical asset is an «-period asset. From the standpoint of period t, the terminal period of an «-period asset is period f+л. This asset pays ау-^Дя_ • in the period that is j periods before the terminal period for j = 0,. . и-l, where yt+n.j > 0 is a random variable, aQ > 0 is a constant, a. > 0, j = 1,…, л -1 are constants, and Я is a constant that indexes the variability of future payoffs. Thus, for instance, in the terminal period, the asset pays a0yf+n; and in period /+1 the asset pays an_Ay^x.
The canonical asset introduced here includes equities and fixed-income securities of all maturities. Fixed-income securities such as bonds and bills are represented by Я = 0 which implies that the payoff in period t+n-j is the known amount a}. A standard coupon bond with face value F and coupon d is represented by a0 =F + d, and a\ =• • • = Д„-1 = d > 0; in this formulation, a pure discount bond is represented by d = 0. Securities with risky payoffs have nonzero values of Я.
For instance, in the Lucas (1978) fruit-tree model, the dividend (per capita) on unlevered equity equals consumption per capita Cr In terms of our canonical asset, unlevered equity in the Lucas model is an infinite-period asset that pays Ct in period t. Using the notation for canonical assets, n = oo, a. = 1 for all j > 0, yt s Ct, and Я = 1. As discussed in section IV, levered equity can be represented by values of Я greater than one.
The benchmark level of consumption, v, tends to grow over time as the standard of living in a country rises. Specifically,
where G > 1 and 0 < у < 1 for i = 0, 1, and 2. The dependence of the benchmark level of consumption on current and lagged aggregate consumption per capita, C, and CM, is a generalization of the “catching up with the Joneses” specification of utility introduced in Abel (1990), and captures the notion that the benchmark level of consumption is an increasing (homogeneous) function of current and recent levels of consumption per capita.
The special case with у > 0 and у = y2 = 0 corresponds the simple formulation of catching up with the Joneses in Abel (1990). The case with у > 0 and / = уг = 0 corresponds to Gali’s (1994) specification of consumption externalities. The dependence of the benchmark level of consumption on Gl allows for the possibility that the benchmark level of consumption grows simply with the passage of time comments.
Section I develops the model with catching up with the Joneses preferences and introduces the canonical asset. Asset pricing is discussed in Section II which includes a discussion of risk premia and term premia without restricting the distribution of growth rates. Beginning in Section III, I assume that growth rates are lognormal more.
The interpretation of the parameter Я as a measure of leverage is discussed in Section IV. In Section V, I derive closed-form expressions for the means and variances of the riskless rate and the rate of return on equity. I calibrate the model in Section VI and develop the algorithm for choosing parameter values that allow the model’s predictions of the unconditional means and variances of the riskless rate and the rate of return on levered equity to match the corresponding empirical moments. Concluding remarks are presented in Section VII.
The catching up with the Joneses feature of the utility function was originally introduced to help account for the high average value of the equity premium observed empirically. However, when this form of the utility function was specified to imply a realistic value of the equity premium in Abel (1990), the model produced a riskless rate of return that was far too volatile.
Campbell and Cochrane (1994) developed a form of catching up with the Joneses preferences that yielded, as in actual data, a large equity premium and low variability of the riskless rate. They achieved this low variability of the riskless rate by specifying a complicated recursive function for the determination of the benchmark level of consumption. Here I adopt a simpler formulation of the benchmark level of consumption that produces, with the inclusion of leverage, low variability of the riskless rate along with a large equity premium.
The average annual rate of return on equity exceeds the average annual rate of return on short-term riskless bills by several hundred basis. There are two components to this equity premium: a risk premium and a term premium. The risk premium reflects the fact that equity is claim on stochastic payoffs, whereas a short-term riskless bill is a fixed-income security that is a claim on a known payoff. The term premium reflects the longer maturity of equity relative to short-term bills fully.
The literature on the equity premium has typically focused on the overall equity premium rather than on the separate components. In the most influential quantitative application of the Lucas (1978) fruit-tree model of asset pricing, Mehra and Prescott (1985) found that this model was incapable of producing an equity premium of more than 35 basis points when they confined attention to parameter values that they deemed plausible.
With such a small overall equity premium implied by the model, it did not much matter how much was a risk premium and how much was a term premium. However, subsequent research has produced models that can account for an equity premium of several hundred basis points per year. I will show in this paper how the equity premium in a general equilibrium model can be decomposed into a risk premium and a term premium. This decomposition provides helpful insights about the source of the equity premium and provides guidance in choosing appropriate parameter values.
The household optimization problem is subject to the constraint that leisure not exceed the endowment of time (equation (2)). For those h iseholds who would violate the constraint, the model calculates shadow wage rates at which they exactly consume their full-time endowment.
The household’s budget constraint is kinked due to the tax deductions applied against wage income. A household with wage income below the deduction level faces marginal and average tax rates equal to zero. A household with wage income above the deduction level faces positive marginal and average tax rates. Due to the discontinuity of the marginal tax rates, it may be optimal for some households to locate exactly at the kink.
Our algorithm deals with this problem as follows. We identify households that choose to locate at the kink by evaluating their leisure choice and corresponding wage income above and below the kink. We then calculate a shadow marginal tax rate from the first-order conditions that puts those households exactly at the kink. This procedure generates optimal forward-looking leisure and consumption choices for all periods of life.